Cell decomposition and dual boundary complexes of character varieties

TL;DR

This paper proves the weak geometric P=W conjecture for certain generic character varieties by decomposing these varieties into simpler parts and analyzing their boundary complexes.

Contribution

It establishes the weak geometric P=W conjecture for very generic $GL_n(C)$-character varieties using an advanced cell decomposition and motivic cohomology techniques.

Findings

Proves the weak geometric P=W conjecture for very generic character varieties.

Develops a refined cell decomposition theorem for character varieties.

Provides a motivic characterization of the cohomology of boundary complexes.

Abstract

The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson asserts that for any smooth Betti moduli space $\mathcal{M}_B$ of complex dimension $d$ over a punctured Riemann surface, the dual boundary complex $\mathbb{D}\partial\mathcal{M}_B$ is homotopy equivalent to a $(d-1)$-dimensional sphere. Here, we consider $\mathcal{M}_B$ as a generic $GL_n(\mathbb{C})$-character variety defined on a Riemann surface of genus $g$, with local monodromies specified by generic semisimple conjugacy classes at $k$ punctures. In this article, we establish the weak geometric P=W conjecture for all \emph{very generic} $\mathcal{M}_B$ in the sense that at least one conjugacy class is regular semisimple. A crucial step is to establish a stronger form of A. Mellit's cell decomposition theorem, i.e. we decompose $\mathcal{M}_B$ (without passing to a vector bundle) into locally closed subvarieties of the form $(\mathbb{C}^{\times})^{d-2b}\times\mathcal{A}$, where $\mathcal{A}$ is stably isomorphic to $\mathbb{C}^b$. A second ingredient involves a motivic characterization of the integral cohomology of dual boundary complexes developed in a subsequent article [Su24]. Following C. Simpson's strategy, the proof is now an inductive computation of the dual boundary complexes from such a cell decomposition.